mirror of
https://github.com/freebsd/freebsd-src
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I'm happy to finally commit stephen@'s implementations of cacos,
cacosh, casin, casinh, catan, and catanh. Thanks to stephen@ and bde@ for working on these. Submitted by: stephen@ Reviewed by: bde
This commit is contained in:
David Schultz
parent
6bba248bee
commit
e4afa19c33
Notes:
svn2git
2020-12-20 02:59:44 +00:00
svn path=/head/; revision=251121
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@ -63,9 +63,21 @@ __BEGIN_DECLS
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double cabs(double complex);
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float cabsf(float complex);
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long double cabsl(long double complex);
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double complex cacos(double complex);
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float complex cacosf(float complex);
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double complex cacosh(double complex);
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float complex cacoshf(float complex);
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double carg(double complex);
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float cargf(float complex);
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long double cargl(long double complex);
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double complex casin(double complex);
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float complex casinf(float complex);
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double complex casinh(double complex);
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float complex casinhf(float complex);
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double complex catan(double complex);
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float complex catanf(float complex);
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double complex catanh(double complex);
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float complex catanhf(float complex);
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double complex ccos(double complex);
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float complex ccosf(float complex);
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double complex ccosh(double complex);
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@ -105,7 +105,8 @@ COMMON_SRCS+= e_acosl.c e_asinl.c e_atan2l.c e_fmodl.c \
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.endif
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# C99 complex functions
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COMMON_SRCS+= s_ccosh.c s_ccoshf.c s_cexp.c s_cexpf.c \
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COMMON_SRCS+= catrig.c catrigf.c \
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s_ccosh.c s_ccoshf.c s_cexp.c s_cexpf.c \
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s_cimag.c s_cimagf.c s_cimagl.c \
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s_conj.c s_conjf.c s_conjl.c \
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s_cproj.c s_cprojf.c s_creal.c s_crealf.c s_creall.c \
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@ -126,7 +127,7 @@ SRCS= ${COMMON_SRCS} ${ARCH_SRCS}
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INCS+= fenv.h math.h
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MAN= acos.3 acosh.3 asin.3 asinh.3 atan.3 atan2.3 atanh.3 \
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ceil.3 ccos.3 ccosh.3 cexp.3 \
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ceil.3 cacos.3 ccos.3 ccosh.3 cexp.3 \
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cimag.3 copysign.3 cos.3 cosh.3 csqrt.3 erf.3 exp.3 fabs.3 fdim.3 \
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feclearexcept.3 feenableexcept.3 fegetenv.3 \
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fegetround.3 fenv.3 floor.3 \
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@ -144,6 +145,9 @@ MLINKS+=atan.3 atanf.3 atan.3 atanl.3
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MLINKS+=atanh.3 atanhf.3
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MLINKS+=atan2.3 atan2f.3 atan2.3 atan2l.3 \
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atan2.3 carg.3 atan2.3 cargf.3 atan2.3 cargl.3
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MLINKS+=cacos.3 cacosf.3 cacos.3 cacosh.3 cacos.3 cacoshf.3 \
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cacos.3 casin.3 cacos.3 casinf.3 cacos.3 casinh.3 cacos.3 casinhf.3 \
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cacos.3 catan.3 cacos.3 catanf.3 cacos.3 catanh.3 cacos.3 catanhf.3
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MLINKS+=ccos.3 ccosf.3 ccos.3 csin.3 ccos.3 csinf.3 ccos.3 ctan.3 ccos.3 ctanf.3
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MLINKS+=ccosh.3 ccoshf.3 ccosh.3 csinh.3 ccosh.3 csinhf.3 \
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ccosh.3 ctanh.3 ccosh.3 ctanhf.3
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@ -237,6 +237,18 @@ FBSD_1.3 {
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fegetround;
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fesetround;
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fesetenv;
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cacos;
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cacosf;
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cacosh;
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cacoshf;
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casin;
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casinf;
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casinh;
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casinhf;
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catan;
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catanf;
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catanh;
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catanhf;
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csin;
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csinf;
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csinh;
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128
lib/msun/man/cacos.3
Normal file
128
lib/msun/man/cacos.3
Normal file
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@ -0,0 +1,128 @@
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.\" Copyright (c) 2013 David Schultz <das@FreeBSD.org>
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.\" All rights reserved.
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.\"
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.\" Redistribution and use in source and binary forms, with or without
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.\" modification, are permitted provided that the following conditions
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.\" are met:
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.\" 1. Redistributions of source code must retain the above copyright
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.\" notice, this list of conditions and the following disclaimer.
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.\" 2. Redistributions in binary form must reproduce the above copyright
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.\" notice, this list of conditions and the following disclaimer in the
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.\" documentation and/or other materials provided with the distribution.
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.\"
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.\" THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
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.\" ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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.\" IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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.\" ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
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.\" FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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.\" DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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.\" OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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.\" HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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.\" LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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.\" OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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.\" SUCH DAMAGE.
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.\"
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.\" $FreeBSD$
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.\"
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.Dd May 27, 2013
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.Dt CACOS 3
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.Os
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.Sh NAME
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.Nm cacos ,
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.Nm cacosf ,
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.Nm cacosh ,
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.Nm cacoshf ,
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.Nm casin ,
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.Nm casinf
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.Nm casinh ,
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.Nm casinhf
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.Nm catan ,
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.Nm catanf
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.Nm catanh ,
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.Nm catanhf
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.Nd complex arc trigonometric and hyperbolic functions
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.Sh LIBRARY
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.Lb libm
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.Sh SYNOPSIS
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.In complex.h
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.Ft double complex
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.Fn cacos "double complex z"
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.Ft float complex
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.Fn cacosf "float complex z"
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.Ft double complex
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.Fn cacosh "double complex z"
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.Ft float complex
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.Fn cacoshf "float complex z"
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.Ft double complex
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.Fn casin "double complex z"
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.Ft float complex
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.Fn casinf "float complex z"
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.Ft double complex
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.Fn casinh "double complex z"
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.Ft float complex
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.Fn casinhf "float complex z"
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.Ft double complex
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.Fn catan "double complex z"
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.Ft float complex
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.Fn catanf "float complex z"
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.Ft double complex
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.Fn catanh "double complex z"
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.Ft float complex
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.Fn catanhf "float complex z"
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.Sh DESCRIPTION
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The
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.Fn cacos ,
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.Fn casin ,
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and
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.Fn catan
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functions compute the principal value of the inverse cosine, sine,
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and tangent of the complex number
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.Fa z ,
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respectively.
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The
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.Fn cacosh ,
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.Fn casinh ,
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and
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.Fn catanh
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functions compute the principal value of the inverse hyperbolic
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cosine, sine, and tangent.
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The
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.Fn cacosf ,
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.Fn casinf ,
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.Fn catanf
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.Fn cacoshf ,
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.Fn casinhf ,
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and
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.Fn catanhf
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functions perform the same operations in
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.Fa float
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precision.
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.Pp
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.ie '\*[.T]'utf8'
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. ds Un \[cu]
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.el
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. ds Un U
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.
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There is no universal convention for defining the principal values of
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these functions. The following table gives the branch cuts, and the
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corresponding ranges for the return values, adopted by the C language.
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.Bl -column ".Sy Function" ".Sy (-\*(If*I, -I) \*(Un (I, \*(If*I)" ".Sy [-\*(Pi/2*I, \*(Pi/2*I]"
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.It Sy Function Ta Sy Branch Cut(s) Ta Sy Range
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.It cacos Ta (-\*(If, -1) \*(Un (1, \*(If) Ta [0, \*(Pi]
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.It casin Ta (-\*(If, -1) \*(Un (1, \*(If) Ta [-\*(Pi/2, \*(Pi/2]
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.It catan Ta (-\*(If*I, -i) \*(Un (I, \*(If*I) Ta [-\*(Pi/2, \*(Pi/2]
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.It cacosh Ta (-\*(If, 1) Ta [-\*(Pi*I, \*(Pi*I]
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.It casinh Ta (-\*(If*I, -i) \*(Un (I, \*(If*I) Ta [-\*(Pi/2*I, \*(Pi/2*I]
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.It catanh Ta (-\*(If, -1) \*(Un (1, \*(If) Ta [-\*(Pi/2*I, \*(Pi/2*I]
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.El
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.Sh SEE ALSO
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.Xr ccos 3 ,
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.Xr ccosh 3 ,
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.Xr complex 3 ,
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.Xr cos 3 ,
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.Xr math 3 ,
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.Xr sin 3 ,
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.Xr tan 3
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.Sh STANDARDS
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These functions conform to
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.St -isoC-99 .
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@ -69,6 +69,7 @@ functions perform the same operations in
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.Fa float
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precision.
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.Sh SEE ALSO
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.Xr cacos 3 ,
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.Xr ccosh 3 ,
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.Xr complex 3 ,
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.Xr cos 3 ,
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@ -69,6 +69,7 @@ functions perform the same operations in
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.Fa float
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precision.
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.Sh SEE ALSO
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.Xr cacosh 3 ,
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.Xr ccos 3 ,
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.Xr complex 3 ,
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.Xr cosh 3 ,
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@ -89,6 +89,12 @@ creal compute the real part
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.\" Section 7.3.5-6 of ISO C99 standard
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.Ss Trigonometric and Hyperbolic Functions
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.Cl
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cacos arc cosine
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cacosh arc hyperbolic cosine
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casin arc sine
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casinh arc hyperbolic sine
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catan arc tangent
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catanh arc hyperbolic tangent
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ccos cosine
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ccosh hyperbolic cosine
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csin sine
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@ -111,20 +117,8 @@ The
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functions described here conform to
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.St -isoC-99 .
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.Sh BUGS
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The inverse trigonometric and hyperbolic functions
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.Fn cacos ,
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.Fn cacosh ,
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.Fn casin ,
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.Fn casinh ,
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.Fn catan ,
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and
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.Fn catanh
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are not implemented.
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.Pp
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The logarithmic functions
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.Fn clog
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are not implemented.
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.Pp
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The power functions
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and the power functions
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.Fn cpow
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are not implemented.
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643
lib/msun/src/catrig.c
Normal file
643
lib/msun/src/catrig.c
Normal file
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@ -0,0 +1,643 @@
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/*-
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* Copyright (c) 2012 Stephen Montgomery-Smith <stephen@FreeBSD.ORG>
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* All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* 1. Redistributions of source code must retain the above copyright
|
||||
* notice, this list of conditions and the following disclaimer.
|
||||
* 2. Redistributions in binary form must reproduce the above copyright
|
||||
* notice, this list of conditions and the following disclaimer in the
|
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* documentation and/or other materials provided with the distribution.
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*
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* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
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* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
|
||||
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
|
||||
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
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||||
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
|
||||
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
|
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* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
|
||||
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
|
||||
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
|
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* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
|
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* SUCH DAMAGE.
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*/
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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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#include <complex.h>
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#include <float.h>
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#include "math.h"
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#include "math_private.h"
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#undef isinf
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#define isinf(x) (fabs(x) == INFINITY)
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#undef isnan
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#define isnan(x) ((x) != (x))
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#define raise_inexact() do { volatile float junk = 1 + tiny; } while(0)
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#undef signbit
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#define signbit(x) (__builtin_signbit(x))
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/* We need that DBL_EPSILON^2/128 is larger than FOUR_SQRT_MIN. */
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static const double
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A_crossover = 10, /* Hull et al suggest 1.5, but 10 works better */
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B_crossover = 0.6417, /* suggested by Hull et al */
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FOUR_SQRT_MIN = 0x1p-509, /* >= 4 * sqrt(DBL_MIN) */
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QUARTER_SQRT_MAX = 0x1p509, /* <= sqrt(DBL_MAX) / 4 */
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m_e = 2.7182818284590452e0, /* 0x15bf0a8b145769.0p-51 */
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m_ln2 = 6.9314718055994531e-1, /* 0x162e42fefa39ef.0p-53 */
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pio2_hi = 1.5707963267948966e0, /* 0x1921fb54442d18.0p-52 */
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RECIP_EPSILON = 1 / DBL_EPSILON,
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SQRT_3_EPSILON = 2.5809568279517849e-8, /* 0x1bb67ae8584caa.0p-78 */
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SQRT_6_EPSILON = 3.6500241499888571e-8, /* 0x13988e1409212e.0p-77 */
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SQRT_MIN = 0x1p-511; /* >= sqrt(DBL_MIN) */
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static const volatile double
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pio2_lo = 6.1232339957367659e-17; /* 0x11a62633145c07.0p-106 */
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static const volatile float
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tiny = 0x1p-100;
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static double complex clog_for_large_values(double complex z);
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/*
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* Testing indicates that all these functions are accurate up to 4 ULP.
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* The functions casin(h) and cacos(h) are about 2.5 times slower than asinh.
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* The functions catan(h) are a little under 2 times slower than atanh.
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*
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* The code for casinh, casin, cacos, and cacosh comes first. The code is
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* rather complicated, and the four functions are highly interdependent.
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*
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* The code for catanh and catan comes at the end. It is much simpler than
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* the other functions, and the code for these can be disconnected from the
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* rest of the code.
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*/
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/*
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* ================================
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* | casinh, casin, cacos, cacosh |
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* ================================
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*/
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/*
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* The algorithm is very close to that in "Implementing the complex arcsine
|
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* and arccosine functions using exception handling" by T. E. Hull, Thomas F.
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* Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on
|
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* Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335,
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* http://dl.acm.org/citation.cfm?id=275324.
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*
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* Throughout we use the convention z = x + I*y.
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*
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* casinh(z) = sign(x)*log(A+sqrt(A*A-1)) + I*asin(B)
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* where
|
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* A = (|z+I| + |z-I|) / 2
|
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* B = (|z+I| - |z-I|) / 2 = y/A
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*
|
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* These formulas become numerically unstable:
|
||||
* (a) for Re(casinh(z)) when z is close to the line segment [-I, I] (that
|
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* is, Re(casinh(z)) is close to 0);
|
||||
* (b) for Im(casinh(z)) when z is close to either of the intervals
|
||||
* [I, I*infinity) or (-I*infinity, -I] (that is, |Im(casinh(z))| is
|
||||
* close to PI/2).
|
||||
*
|
||||
* These numerical problems are overcome by defining
|
||||
* f(a, b) = (hypot(a, b) - b) / 2 = a*a / (hypot(a, b) + b) / 2
|
||||
* Then if A < A_crossover, we use
|
||||
* log(A + sqrt(A*A-1)) = log1p((A-1) + sqrt((A-1)*(A+1)))
|
||||
* A-1 = f(x, 1+y) + f(x, 1-y)
|
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* and if B > B_crossover, we use
|
||||
* asin(B) = atan2(y, sqrt(A*A - y*y)) = atan2(y, sqrt((A+y)*(A-y)))
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* A-y = f(x, y+1) + f(x, y-1)
|
||||
* where without loss of generality we have assumed that x and y are
|
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* non-negative.
|
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*
|
||||
* Much of the difficulty comes because the intermediate computations may
|
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* produce overflows or underflows. This is dealt with in the paper by Hull
|
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* et al by using exception handling. We do this by detecting when
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* computations risk underflow or overflow. The hardest part is handling the
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* underflows when computing f(a, b).
|
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*
|
||||
* Note that the function f(a, b) does not appear explicitly in the paper by
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* Hull et al, but the idea may be found on pages 308 and 309. Introducing the
|
||||
* function f(a, b) allows us to concentrate many of the clever tricks in this
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* paper into one function.
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*/
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/*
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* Function f(a, b, hypot_a_b) = (hypot(a, b) - b) / 2.
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* Pass hypot(a, b) as the third argument.
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||||
*/
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static inline double
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f(double a, double b, double hypot_a_b)
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{
|
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if (b < 0)
|
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return ((hypot_a_b - b) / 2);
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||||
if (b == 0)
|
||||
return (a / 2);
|
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return (a * a / (hypot_a_b + b) / 2);
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}
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|
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/*
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* All the hard work is contained in this function.
|
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* x and y are assumed positive or zero, and less than RECIP_EPSILON.
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* Upon return:
|
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* rx = Re(casinh(z)) = -Im(cacos(y + I*x)).
|
||||
* B_is_usable is set to 1 if the value of B is usable.
|
||||
* If B_is_usable is set to 0, sqrt_A2my2 = sqrt(A*A - y*y), and new_y = y.
|
||||
* If returning sqrt_A2my2 has potential to result in an underflow, it is
|
||||
* rescaled, and new_y is similarly rescaled.
|
||||
*/
|
||||
static inline void
|
||||
do_hard_work(double x, double y, double *rx, int *B_is_usable, double *B,
|
||||
double *sqrt_A2my2, double *new_y)
|
||||
{
|
||||
double R, S, A; /* A, B, R, and S are as in Hull et al. */
|
||||
double Am1, Amy; /* A-1, A-y. */
|
||||
|
||||
R = hypot(x, y + 1); /* |z+I| */
|
||||
S = hypot(x, y - 1); /* |z-I| */
|
||||
|
||||
/* A = (|z+I| + |z-I|) / 2 */
|
||||
A = (R + S) / 2;
|
||||
/*
|
||||
* Mathematically A >= 1. There is a small chance that this will not
|
||||
* be so because of rounding errors. So we will make certain it is
|
||||
* so.
|
||||
*/
|
||||
if (A < 1)
|
||||
A = 1;
|
||||
|
||||
if (A < A_crossover) {
|
||||
/*
|
||||
* Am1 = fp + fm, where fp = f(x, 1+y), and fm = f(x, 1-y).
|
||||
* rx = log1p(Am1 + sqrt(Am1*(A+1)))
|
||||
*/
|
||||
if (y == 1 && x < DBL_EPSILON*DBL_EPSILON / 128) {
|
||||
/*
|
||||
* fp is of order x^2, and fm = x/2.
|
||||
* A = 1 (inexactly).
|
||||
*/
|
||||
*rx = sqrt(x);
|
||||
} else if (x >= DBL_EPSILON * fabs(y - 1)) {
|
||||
/*
|
||||
* Underflow will not occur because
|
||||
* x >= DBL_EPSILON^2/128 >= FOUR_SQRT_MIN
|
||||
*/
|
||||
Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
|
||||
*rx = log1p(Am1 + sqrt(Am1 * (A + 1)));
|
||||
} else if (y < 1) {
|
||||
/*
|
||||
* fp = x*x/(1+y)/4, fm = x*x/(1-y)/4, and
|
||||
* A = 1 (inexactly).
|
||||
*/
|
||||
*rx = x / sqrt((1 - y) * (1 + y));
|
||||
} else /* if (y > 1) */ {
|
||||
/*
|
||||
* A-1 = y-1 (inexactly).
|
||||
*/
|
||||
*rx = log1p((y - 1) + sqrt((y - 1) * (y + 1)));
|
||||
}
|
||||
} else {
|
||||
*rx = log(A + sqrt(A * A - 1));
|
||||
}
|
||||
|
||||
*new_y = y;
|
||||
|
||||
if (y < FOUR_SQRT_MIN) {
|
||||
/*
|
||||
* Avoid a possible underflow caused by y/A. For casinh this
|
||||
* would be legitimate, but will be picked up by invoking atan2
|
||||
* later on. For cacos this would not be legitimate.
|
||||
*/
|
||||
*B_is_usable = 0;
|
||||
*sqrt_A2my2 = A * (2 / DBL_EPSILON);
|
||||
*new_y = y * (2 / DBL_EPSILON);
|
||||
return;
|
||||
}
|
||||
|
||||
/* B = (|z+I| - |z-I|) / 2 = y/A */
|
||||
*B = y / A;
|
||||
*B_is_usable = 1;
|
||||
|
||||
if (*B > B_crossover) {
|
||||
*B_is_usable = 0;
|
||||
/*
|
||||
* Amy = fp + fm, where fp = f(x, y+1), and fm = f(x, y-1).
|
||||
* sqrt_A2my2 = sqrt(Amy*(A+y))
|
||||
*/
|
||||
if (y == 1 && x < DBL_EPSILON / 128) {
|
||||
/*
|
||||
* fp is of order x^2, and fm = x/2.
|
||||
* A = 1 (inexactly).
|
||||
*/
|
||||
*sqrt_A2my2 = sqrt(x) * sqrt((A + y) / 2);
|
||||
} else if (x >= DBL_EPSILON * fabs(y - 1)) {
|
||||
/*
|
||||
* Underflow will not occur because
|
||||
* x >= DBL_EPSILON/128 >= FOUR_SQRT_MIN
|
||||
* and
|
||||
* x >= DBL_EPSILON^2 >= FOUR_SQRT_MIN
|
||||
*/
|
||||
Amy = f(x, y + 1, R) + f(x, y - 1, S);
|
||||
*sqrt_A2my2 = sqrt(Amy * (A + y));
|
||||
} else if (y > 1) {
|
||||
/*
|
||||
* fp = x*x/(y+1)/4, fm = x*x/(y-1)/4, and
|
||||
* A = y (inexactly).
|
||||
*
|
||||
* y < RECIP_EPSILON. So the following
|
||||
* scaling should avoid any underflow problems.
|
||||
*/
|
||||
*sqrt_A2my2 = x * (4 / DBL_EPSILON / DBL_EPSILON) * y /
|
||||
sqrt((y + 1) * (y - 1));
|
||||
*new_y = y * (4 / DBL_EPSILON / DBL_EPSILON);
|
||||
} else /* if (y < 1) */ {
|
||||
/*
|
||||
* fm = 1-y >= DBL_EPSILON, fp is of order x^2, and
|
||||
* A = 1 (inexactly).
|
||||
*/
|
||||
*sqrt_A2my2 = sqrt((1 - y) * (1 + y));
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
/*
|
||||
* casinh(z) = z + O(z^3) as z -> 0
|
||||
*
|
||||
* casinh(z) = sign(x)*clog(sign(x)*z) + O(1/z^2) as z -> infinity
|
||||
* The above formula works for the imaginary part as well, because
|
||||
* Im(casinh(z)) = sign(x)*atan2(sign(x)*y, fabs(x)) + O(y/z^3)
|
||||
* as z -> infinity, uniformly in y
|
||||
*/
|
||||
double complex
|
||||
casinh(double complex z)
|
||||
{
|
||||
double x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
|
||||
int B_is_usable;
|
||||
double complex w;
|
||||
|
||||
x = creal(z);
|
||||
y = cimag(z);
|
||||
ax = fabs(x);
|
||||
ay = fabs(y);
|
||||
|
||||
if (isnan(x) || isnan(y)) {
|
||||
/* casinh(+-Inf + I*NaN) = +-Inf + I*NaN */
|
||||
if (isinf(x))
|
||||
return (cpack(x, y + y));
|
||||
/* casinh(NaN + I*+-Inf) = opt(+-)Inf + I*NaN */
|
||||
if (isinf(y))
|
||||
return (cpack(y, x + x));
|
||||
/* casinh(NaN + I*0) = NaN + I*0 */
|
||||
if (y == 0)
|
||||
return (cpack(x + x, y));
|
||||
/*
|
||||
* All other cases involving NaN return NaN + I*NaN.
|
||||
* C99 leaves it optional whether to raise invalid if one of
|
||||
* the arguments is not NaN, so we opt not to raise it.
|
||||
*/
|
||||
/* Bruce Evans tells me this is the way to do this: */
|
||||
return (cpack(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
|
||||
}
|
||||
|
||||
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
|
||||
/* clog...() will raise inexact unless x or y is infinite. */
|
||||
if (signbit(x) == 0)
|
||||
w = clog_for_large_values(z) + m_ln2;
|
||||
else
|
||||
w = clog_for_large_values(-z) + m_ln2;
|
||||
return (cpack(copysign(creal(w), x), copysign(cimag(w), y)));
|
||||
}
|
||||
|
||||
/* Avoid spuriously raising inexact for z = 0. */
|
||||
if (x == 0 && y == 0)
|
||||
return (z);
|
||||
|
||||
/* All remaining cases are inexact. */
|
||||
raise_inexact();
|
||||
|
||||
if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
|
||||
return (z);
|
||||
|
||||
do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
|
||||
if (B_is_usable)
|
||||
ry = asin(B);
|
||||
else
|
||||
ry = atan2(new_y, sqrt_A2my2);
|
||||
return (cpack(copysign(rx, x), copysign(ry, y)));
|
||||
}
|
||||
|
||||
/*
|
||||
* casin(z) = reverse(casinh(reverse(z)))
|
||||
* where reverse(x + I*y) = y + I*x = I*conj(z).
|
||||
*/
|
||||
double complex
|
||||
casin(double complex z)
|
||||
{
|
||||
double complex w = casinh(cpack(cimag(z), creal(z)));
|
||||
return (cpack(cimag(w), creal(w)));
|
||||
}
|
||||
|
||||
/*
|
||||
* cacos(z) = PI/2 - casin(z)
|
||||
* but do the computation carefully so cacos(z) is accurate when z is
|
||||
* close to 1.
|
||||
*
|
||||
* cacos(z) = PI/2 - z + O(z^3) as z -> 0
|
||||
*
|
||||
* cacos(z) = -sign(y)*I*clog(z) + O(1/z^2) as z -> infinity
|
||||
* The above formula works for the real part as well, because
|
||||
* Re(cacos(z)) = atan2(fabs(y), x) + O(y/z^3)
|
||||
* as z -> infinity, uniformly in y
|
||||
*/
|
||||
double complex
|
||||
cacos(double complex z)
|
||||
{
|
||||
double x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
|
||||
int sx, sy;
|
||||
int B_is_usable;
|
||||
double complex w;
|
||||
|
||||
x = creal(z);
|
||||
y = cimag(z);
|
||||
sx = signbit(x);
|
||||
sy = signbit(y);
|
||||
ax = fabs(x);
|
||||
ay = fabs(y);
|
||||
|
||||
if (isnan(x) || isnan(y)) {
|
||||
/* cacos(+-Inf + I*NaN) = NaN + I*opt(-)Inf */
|
||||
if (isinf(x))
|
||||
return (cpack(y + y, -INFINITY));
|
||||
/* cacos(NaN + I*+-Inf) = NaN + I*-+Inf */
|
||||
if (isinf(y))
|
||||
return (cpack(x + x, -y));
|
||||
/* cacos(0 + I*NaN) = PI/2 + I*NaN with inexact */
|
||||
if (x == 0)
|
||||
return (cpack(pio2_hi + pio2_lo, y + y));
|
||||
/*
|
||||
* All other cases involving NaN return NaN + I*NaN.
|
||||
* C99 leaves it optional whether to raise invalid if one of
|
||||
* the arguments is not NaN, so we opt not to raise it.
|
||||
*/
|
||||
return (cpack(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
|
||||
}
|
||||
|
||||
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
|
||||
/* clog...() will raise inexact unless x or y is infinite. */
|
||||
w = clog_for_large_values(z);
|
||||
rx = fabs(cimag(w));
|
||||
ry = creal(w) + m_ln2;
|
||||
if (sy == 0)
|
||||
ry = -ry;
|
||||
return (cpack(rx, ry));
|
||||
}
|
||||
|
||||
/* Avoid spuriously raising inexact for z = 1. */
|
||||
if (x == 1 && y == 0)
|
||||
return (cpack(0, -y));
|
||||
|
||||
/* All remaining cases are inexact. */
|
||||
raise_inexact();
|
||||
|
||||
if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON/4)
|
||||
return (cpack(pio2_hi - (x - pio2_lo), -y));
|
||||
|
||||
do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
|
||||
if (B_is_usable) {
|
||||
if (sx==0)
|
||||
rx = acos(B);
|
||||
else
|
||||
rx = acos(-B);
|
||||
} else {
|
||||
if (sx==0)
|
||||
rx = atan2(sqrt_A2mx2, new_x);
|
||||
else
|
||||
rx = atan2(sqrt_A2mx2, -new_x);
|
||||
}
|
||||
if (sy == 0)
|
||||
ry = -ry;
|
||||
return (cpack(rx, ry));
|
||||
}
|
||||
|
||||
/*
|
||||
* cacosh(z) = I*cacos(z) or -I*cacos(z)
|
||||
* where the sign is chosen so Re(cacosh(z)) >= 0.
|
||||
*/
|
||||
double complex
|
||||
cacosh(double complex z)
|
||||
{
|
||||
double complex w;
|
||||
double rx, ry;
|
||||
|
||||
w = cacos(z);
|
||||
rx = creal(w);
|
||||
ry = cimag(w);
|
||||
/* cacosh(NaN + I*NaN) = NaN + I*NaN */
|
||||
if (isnan(rx) && isnan(ry))
|
||||
return (cpack(ry, rx));
|
||||
/* cacosh(NaN + I*+-Inf) = +Inf + I*NaN */
|
||||
/* cacosh(+-Inf + I*NaN) = +Inf + I*NaN */
|
||||
if (isnan(rx))
|
||||
return (cpack(fabs(ry), rx));
|
||||
/* cacosh(0 + I*NaN) = NaN + I*NaN */
|
||||
if (isnan(ry))
|
||||
return (cpack(ry, ry));
|
||||
return (cpack(fabs(ry), copysign(rx, cimag(z))));
|
||||
}
|
||||
|
||||
/*
|
||||
* Optimized version of clog() for |z| finite and larger than ~RECIP_EPSILON.
|
||||
*/
|
||||
static double complex
|
||||
clog_for_large_values(double complex z)
|
||||
{
|
||||
double x, y;
|
||||
double ax, ay, t;
|
||||
|
||||
x = creal(z);
|
||||
y = cimag(z);
|
||||
ax = fabs(x);
|
||||
ay = fabs(y);
|
||||
if (ax < ay) {
|
||||
t = ax;
|
||||
ax = ay;
|
||||
ay = t;
|
||||
}
|
||||
|
||||
/*
|
||||
* Avoid overflow in hypot() when x and y are both very large.
|
||||
* Divide x and y by E, and then add 1 to the logarithm. This depends
|
||||
* on E being larger than sqrt(2).
|
||||
* Dividing by E causes an insignificant loss of accuracy; however
|
||||
* this method is still poor since it is uneccessarily slow.
|
||||
*/
|
||||
if (ax > DBL_MAX / 2)
|
||||
return (cpack(log(hypot(x / m_e, y / m_e)) + 1, atan2(y, x)));
|
||||
|
||||
/*
|
||||
* Avoid overflow when x or y is large. Avoid underflow when x or
|
||||
* y is small.
|
||||
*/
|
||||
if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
|
||||
return (cpack(log(hypot(x, y)), atan2(y, x)));
|
||||
|
||||
return (cpack(log(ax * ax + ay * ay) / 2, atan2(y, x)));
|
||||
}
|
||||
|
||||
/*
|
||||
*=============================================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* =================
|
||||
* | catanh, catan |
|
||||
* =================
|
||||
*/
|
||||
|
||||
/*
|
||||
* sum_squares(x,y) = x*x + y*y (or just x*x if y*y would underflow).
|
||||
* Assumes x*x and y*y will not overflow.
|
||||
* Assumes x and y are finite.
|
||||
* Assumes y is non-negative.
|
||||
* Assumes fabs(x) >= DBL_EPSILON.
|
||||
*/
|
||||
static inline double
|
||||
sum_squares(double x, double y)
|
||||
{
|
||||
|
||||
/* Avoid underflow when y is small. */
|
||||
if (y < SQRT_MIN)
|
||||
return (x * x);
|
||||
return (x * x + y * y);
|
||||
}
|
||||
|
||||
/*
|
||||
* real_part_reciprocal(x, y) = Re(1/(x+I*y)) = x/(x*x + y*y).
|
||||
* Assumes x and y are not NaN, and one of x and y is larger than
|
||||
* RECIP_EPSILON. We avoid unwarranted underflow. It is important to not use
|
||||
* the code creal(1/z), because the imaginary part may produce an unwanted
|
||||
* underflow.
|
||||
* This is only called in a context where inexact is always raised before
|
||||
* the call, so no effort is made to avoid or force inexact.
|
||||
*/
|
||||
static inline double
|
||||
real_part_reciprocal(double x, double y)
|
||||
{
|
||||
double scale;
|
||||
uint32_t hx, hy;
|
||||
int32_t ix, iy;
|
||||
|
||||
/*
|
||||
* This code is inspired by the C99 document n1124.pdf, Section G.5.1,
|
||||
* example 2.
|
||||
*/
|
||||
GET_HIGH_WORD(hx, x);
|
||||
ix = hx & 0x7ff00000;
|
||||
GET_HIGH_WORD(hy, y);
|
||||
iy = hy & 0x7ff00000;
|
||||
#define BIAS (DBL_MAX_EXP - 1)
|
||||
/* XXX more guard digits are useful iff there is extra precision. */
|
||||
#define CUTOFF (DBL_MANT_DIG / 2 + 1) /* just half or 1 guard digit */
|
||||
if (ix - iy >= CUTOFF << 20 || isinf(x))
|
||||
return (1 / x); /* +-Inf -> +-0 is special */
|
||||
if (iy - ix >= CUTOFF << 20)
|
||||
return (x / y / y); /* should avoid double div, but hard */
|
||||
if (ix <= (BIAS + DBL_MAX_EXP / 2 - CUTOFF) << 20)
|
||||
return (x / (x * x + y * y));
|
||||
scale = 1;
|
||||
SET_HIGH_WORD(scale, 0x7ff00000 - ix); /* 2**(1-ilogb(x)) */
|
||||
x *= scale;
|
||||
y *= scale;
|
||||
return (x / (x * x + y * y) * scale);
|
||||
}
|
||||
|
||||
/*
|
||||
* catanh(z) = log((1+z)/(1-z)) / 2
|
||||
* = log1p(4*x / |z-1|^2) / 4
|
||||
* + I * atan2(2*y, (1-x)*(1+x)-y*y) / 2
|
||||
*
|
||||
* catanh(z) = z + O(z^3) as z -> 0
|
||||
*
|
||||
* catanh(z) = 1/z + sign(y)*I*PI/2 + O(1/z^3) as z -> infinity
|
||||
* The above formula works for the real part as well, because
|
||||
* Re(catanh(z)) = x/|z|^2 + O(x/z^4)
|
||||
* as z -> infinity, uniformly in x
|
||||
*/
|
||||
double complex
|
||||
catanh(double complex z)
|
||||
{
|
||||
double x, y, ax, ay, rx, ry;
|
||||
|
||||
x = creal(z);
|
||||
y = cimag(z);
|
||||
ax = fabs(x);
|
||||
ay = fabs(y);
|
||||
|
||||
/* This helps handle many cases. */
|
||||
if (y == 0 && ax <= 1)
|
||||
return (cpack(atanh(x), y));
|
||||
|
||||
/* To ensure the same accuracy as atan(), and to filter out z = 0. */
|
||||
if (x == 0)
|
||||
return (cpack(x, atan(y)));
|
||||
|
||||
if (isnan(x) || isnan(y)) {
|
||||
/* catanh(+-Inf + I*NaN) = +-0 + I*NaN */
|
||||
if (isinf(x))
|
||||
return (cpack(copysign(0, x), y + y));
|
||||
/* catanh(NaN + I*+-Inf) = sign(NaN)0 + I*+-PI/2 */
|
||||
if (isinf(y)) {
|
||||
return (cpack(copysign(0, x),
|
||||
copysign(pio2_hi + pio2_lo, y)));
|
||||
}
|
||||
/*
|
||||
* All other cases involving NaN return NaN + I*NaN.
|
||||
* C99 leaves it optional whether to raise invalid if one of
|
||||
* the arguments is not NaN, so we opt not to raise it.
|
||||
*/
|
||||
return (cpack(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
|
||||
}
|
||||
|
||||
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
|
||||
return (cpack(real_part_reciprocal(x, y),
|
||||
copysign(pio2_hi + pio2_lo, y)));
|
||||
}
|
||||
|
||||
if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
|
||||
/*
|
||||
* z = 0 was filtered out above. All other cases must raise
|
||||
* inexact, but this is the only only that needs to do it
|
||||
* explicitly.
|
||||
*/
|
||||
raise_inexact();
|
||||
return (z);
|
||||
}
|
||||
|
||||
if (ax == 1 && ay < DBL_EPSILON)
|
||||
rx = (log(ay) - m_ln2) / -2;
|
||||
else
|
||||
rx = log1p(4 * ax / sum_squares(ax - 1, ay)) / 4;
|
||||
|
||||
if (ax == 1)
|
||||
ry = atan2(2, -ay) / 2;
|
||||
else if (ay < DBL_EPSILON)
|
||||
ry = atan2(2 * ay, (1 - ax) * (1 + ax)) / 2;
|
||||
else
|
||||
ry = atan2(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
|
||||
|
||||
return (cpack(copysign(rx, x), copysign(ry, y)));
|
||||
}
|
||||
|
||||
/*
|
||||
* catan(z) = reverse(catanh(reverse(z)))
|
||||
* where reverse(x + I*y) = y + I*x = I*conj(z).
|
||||
*/
|
||||
double complex
|
||||
catan(double complex z)
|
||||
{
|
||||
double complex w = catanh(cpack(cimag(z), creal(z)));
|
||||
return (cpack(cimag(w), creal(w)));
|
||||
}
|
||||
388
lib/msun/src/catrigf.c
Normal file
388
lib/msun/src/catrigf.c
Normal file
|
|
@ -0,0 +1,388 @@
|
|||
/*-
|
||||
* Copyright (c) 2012 Stephen Montgomery-Smith <stephen@FreeBSD.ORG>
|
||||
* All rights reserved.
|
||||
*
|
||||
* Redistribution and use in source and binary forms, with or without
|
||||
* modification, are permitted provided that the following conditions
|
||||
* are met:
|
||||
* 1. Redistributions of source code must retain the above copyright
|
||||
* notice, this list of conditions and the following disclaimer.
|
||||
* 2. Redistributions in binary form must reproduce the above copyright
|
||||
* notice, this list of conditions and the following disclaimer in the
|
||||
* documentation and/or other materials provided with the distribution.
|
||||
*
|
||||
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
|
||||
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
|
||||
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
|
||||
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
|
||||
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
|
||||
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
|
||||
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
|
||||
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
|
||||
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
|
||||
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
|
||||
* SUCH DAMAGE.
|
||||
*/
|
||||
|
||||
/*
|
||||
* The algorithm is very close to that in "Implementing the complex arcsine
|
||||
* and arccosine functions using exception handling" by T. E. Hull, Thomas F.
|
||||
* Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on
|
||||
* Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335,
|
||||
* http://dl.acm.org/citation.cfm?id=275324.
|
||||
*
|
||||
* The code for catrig.c contains complete comments.
|
||||
*/
|
||||
|
||||
#include <sys/cdefs.h>
|
||||
__FBSDID("$FreeBSD$");
|
||||
|
||||
#include <complex.h>
|
||||
#include <float.h>
|
||||
|
||||
#include "math.h"
|
||||
#include "math_private.h"
|
||||
|
||||
#undef isinf
|
||||
#define isinf(x) (fabsf(x) == INFINITY)
|
||||
#undef isnan
|
||||
#define isnan(x) ((x) != (x))
|
||||
#define raise_inexact() do { volatile float junk = 1 + tiny; } while(0)
|
||||
#undef signbit
|
||||
#define signbit(x) (__builtin_signbitf(x))
|
||||
|
||||
static const float
|
||||
A_crossover = 10,
|
||||
B_crossover = 0.6417,
|
||||
FOUR_SQRT_MIN = 0x1p-61,
|
||||
QUARTER_SQRT_MAX = 0x1p61,
|
||||
m_e = 2.7182818285e0, /* 0xadf854.0p-22 */
|
||||
m_ln2 = 6.9314718056e-1, /* 0xb17218.0p-24 */
|
||||
pio2_hi = 1.5707962513e0, /* 0xc90fda.0p-23 */
|
||||
RECIP_EPSILON = 1 / FLT_EPSILON,
|
||||
SQRT_3_EPSILON = 5.9801995673e-4, /* 0x9cc471.0p-34 */
|
||||
SQRT_6_EPSILON = 8.4572793338e-4, /* 0xddb3d7.0p-34 */
|
||||
SQRT_MIN = 0x1p-63;
|
||||
|
||||
static const volatile float
|
||||
pio2_lo = 7.5497899549e-8, /* 0xa22169.0p-47 */
|
||||
tiny = 0x1p-100;
|
||||
|
||||
static float complex clog_for_large_values(float complex z);
|
||||
|
||||
static inline float
|
||||
f(float a, float b, float hypot_a_b)
|
||||
{
|
||||
if (b < 0)
|
||||
return ((hypot_a_b - b) / 2);
|
||||
if (b == 0)
|
||||
return (a / 2);
|
||||
return (a * a / (hypot_a_b + b) / 2);
|
||||
}
|
||||
|
||||
static inline void
|
||||
do_hard_work(float x, float y, float *rx, int *B_is_usable, float *B,
|
||||
float *sqrt_A2my2, float *new_y)
|
||||
{
|
||||
float R, S, A;
|
||||
float Am1, Amy;
|
||||
|
||||
R = hypotf(x, y + 1);
|
||||
S = hypotf(x, y - 1);
|
||||
|
||||
A = (R + S) / 2;
|
||||
if (A < 1)
|
||||
A = 1;
|
||||
|
||||
if (A < A_crossover) {
|
||||
if (y == 1 && x < FLT_EPSILON * FLT_EPSILON / 128) {
|
||||
*rx = sqrtf(x);
|
||||
} else if (x >= FLT_EPSILON * fabsf(y - 1)) {
|
||||
Am1 = f(x, 1 + y, R) + f(x, 1 - y, S);
|
||||
*rx = log1pf(Am1 + sqrtf(Am1 * (A + 1)));
|
||||
} else if (y < 1) {
|
||||
*rx = x / sqrtf((1 - y)*(1 + y));
|
||||
} else {
|
||||
*rx = log1pf((y - 1) + sqrtf((y - 1) * (y + 1)));
|
||||
}
|
||||
} else {
|
||||
*rx = logf(A + sqrtf(A * A - 1));
|
||||
}
|
||||
|
||||
*new_y = y;
|
||||
|
||||
if (y < FOUR_SQRT_MIN) {
|
||||
*B_is_usable = 0;
|
||||
*sqrt_A2my2 = A * (2 / FLT_EPSILON);
|
||||
*new_y = y * (2 / FLT_EPSILON);
|
||||
return;
|
||||
}
|
||||
|
||||
*B = y / A;
|
||||
*B_is_usable = 1;
|
||||
|
||||
if (*B > B_crossover) {
|
||||
*B_is_usable = 0;
|
||||
if (y == 1 && x < FLT_EPSILON / 128) {
|
||||
*sqrt_A2my2 = sqrtf(x) * sqrtf((A + y) / 2);
|
||||
} else if (x >= FLT_EPSILON * fabsf(y - 1)) {
|
||||
Amy = f(x, y + 1, R) + f(x, y - 1, S);
|
||||
*sqrt_A2my2 = sqrtf(Amy * (A + y));
|
||||
} else if (y > 1) {
|
||||
*sqrt_A2my2 = x * (4 / FLT_EPSILON / FLT_EPSILON) * y /
|
||||
sqrtf((y + 1) * (y - 1));
|
||||
*new_y = y * (4 / FLT_EPSILON / FLT_EPSILON);
|
||||
} else {
|
||||
*sqrt_A2my2 = sqrtf((1 - y) * (1 + y));
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
float complex
|
||||
casinhf(float complex z)
|
||||
{
|
||||
float x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y;
|
||||
int B_is_usable;
|
||||
float complex w;
|
||||
|
||||
x = crealf(z);
|
||||
y = cimagf(z);
|
||||
ax = fabsf(x);
|
||||
ay = fabsf(y);
|
||||
|
||||
if (isnan(x) || isnan(y)) {
|
||||
if (isinf(x))
|
||||
return (cpackf(x, y + y));
|
||||
if (isinf(y))
|
||||
return (cpackf(y, x + x));
|
||||
if (y == 0)
|
||||
return (cpackf(x + x, y));
|
||||
return (cpackf(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
|
||||
}
|
||||
|
||||
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
|
||||
if (signbit(x) == 0)
|
||||
w = clog_for_large_values(z) + m_ln2;
|
||||
else
|
||||
w = clog_for_large_values(-z) + m_ln2;
|
||||
return (cpackf(copysignf(crealf(w), x),
|
||||
copysignf(cimagf(w), y)));
|
||||
}
|
||||
|
||||
if (x == 0 && y == 0)
|
||||
return (z);
|
||||
|
||||
raise_inexact();
|
||||
|
||||
if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
|
||||
return (z);
|
||||
|
||||
do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y);
|
||||
if (B_is_usable)
|
||||
ry = asinf(B);
|
||||
else
|
||||
ry = atan2f(new_y, sqrt_A2my2);
|
||||
return (cpackf(copysignf(rx, x), copysignf(ry, y)));
|
||||
}
|
||||
|
||||
float complex
|
||||
casinf(float complex z)
|
||||
{
|
||||
float complex w = casinhf(cpackf(cimagf(z), crealf(z)));
|
||||
return (cpackf(cimagf(w), crealf(w)));
|
||||
}
|
||||
|
||||
float complex
|
||||
cacosf(float complex z)
|
||||
{
|
||||
float x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x;
|
||||
int sx, sy;
|
||||
int B_is_usable;
|
||||
float complex w;
|
||||
|
||||
x = crealf(z);
|
||||
y = cimagf(z);
|
||||
sx = signbit(x);
|
||||
sy = signbit(y);
|
||||
ax = fabsf(x);
|
||||
ay = fabsf(y);
|
||||
|
||||
if (isnan(x) || isnan(y)) {
|
||||
if (isinf(x))
|
||||
return (cpackf(y + y, -INFINITY));
|
||||
if (isinf(y))
|
||||
return (cpackf(x + x, -y));
|
||||
if (x == 0) return (cpackf(pio2_hi + pio2_lo, y + y));
|
||||
return (cpackf(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
|
||||
}
|
||||
|
||||
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
|
||||
w = clog_for_large_values(z);
|
||||
rx = fabsf(cimagf(w));
|
||||
ry = crealf(w) + m_ln2;
|
||||
if (sy == 0)
|
||||
ry = -ry;
|
||||
return (cpackf(rx, ry));
|
||||
}
|
||||
|
||||
if (x == 1 && y == 0)
|
||||
return (cpackf(0, -y));
|
||||
|
||||
raise_inexact();
|
||||
|
||||
if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4)
|
||||
return (cpackf(pio2_hi - (x - pio2_lo), -y));
|
||||
|
||||
do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x);
|
||||
if (B_is_usable) {
|
||||
if (sx==0)
|
||||
rx = acosf(B);
|
||||
else
|
||||
rx = acosf(-B);
|
||||
} else {
|
||||
if (sx==0)
|
||||
rx = atan2f(sqrt_A2mx2, new_x);
|
||||
else
|
||||
rx = atan2f(sqrt_A2mx2, -new_x);
|
||||
}
|
||||
if (sy==0)
|
||||
ry = -ry;
|
||||
return (cpackf(rx, ry));
|
||||
}
|
||||
|
||||
float complex
|
||||
cacoshf(float complex z)
|
||||
{
|
||||
float complex w;
|
||||
float rx, ry;
|
||||
|
||||
w = cacosf(z);
|
||||
rx = crealf(w);
|
||||
ry = cimagf(w);
|
||||
if (isnan(rx) && isnan(ry))
|
||||
return (cpackf(ry, rx));
|
||||
if (isnan(rx))
|
||||
return (cpackf(fabsf(ry), rx));
|
||||
if (isnan(ry))
|
||||
return (cpackf(ry, ry));
|
||||
return (cpackf(fabsf(ry), copysignf(rx, cimagf(z))));
|
||||
}
|
||||
|
||||
static float complex
|
||||
clog_for_large_values(float complex z)
|
||||
{
|
||||
float x, y;
|
||||
float ax, ay, t;
|
||||
|
||||
x = crealf(z);
|
||||
y = cimagf(z);
|
||||
ax = fabsf(x);
|
||||
ay = fabsf(y);
|
||||
if (ax < ay) {
|
||||
t = ax;
|
||||
ax = ay;
|
||||
ay = t;
|
||||
}
|
||||
|
||||
if (ax > FLT_MAX / 2) {
|
||||
return (cpackf(logf(hypotf(x / m_e, y / m_e)) + 1,
|
||||
atan2f(y, x)));
|
||||
}
|
||||
|
||||
if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN)
|
||||
return (cpackf(logf(hypotf(x, y)), atan2f(y, x)));
|
||||
|
||||
return (cpackf(logf(ax * ax + ay * ay) / 2, atan2f(y, x)));
|
||||
}
|
||||
|
||||
static inline float
|
||||
sum_squares(float x, float y)
|
||||
{
|
||||
|
||||
if (y < SQRT_MIN)
|
||||
return (x*x);
|
||||
return (x*x + y*y);
|
||||
}
|
||||
|
||||
static inline float
|
||||
real_part_reciprocal(float x, float y)
|
||||
{
|
||||
float scale;
|
||||
uint32_t hx, hy;
|
||||
int32_t ix, iy;
|
||||
|
||||
GET_FLOAT_WORD(hx, x);
|
||||
ix = hx & 0x7f800000;
|
||||
GET_FLOAT_WORD(hy, y);
|
||||
iy = hy & 0x7f800000;
|
||||
#define BIAS (FLT_MAX_EXP - 1)
|
||||
#define CUTOFF (FLT_MANT_DIG / 2 + 1)
|
||||
if (ix - iy >= CUTOFF << 23 || isinf(x))
|
||||
return (1/x);
|
||||
if (iy - ix >= CUTOFF << 23)
|
||||
return (x/y/y);
|
||||
if (ix <= (BIAS + FLT_MAX_EXP / 2 - CUTOFF) << 23)
|
||||
return (x / (x * x + y * y));
|
||||
SET_FLOAT_WORD(scale, 0x7f800000 - ix);
|
||||
x *= scale;
|
||||
y *= scale;
|
||||
return (x / (x * x + y * y) * scale);
|
||||
}
|
||||
|
||||
float complex
|
||||
catanhf(float complex z)
|
||||
{
|
||||
float x, y, ax, ay, rx, ry;
|
||||
|
||||
x = crealf(z);
|
||||
y = cimagf(z);
|
||||
ax = fabsf(x);
|
||||
ay = fabsf(y);
|
||||
|
||||
if (y == 0 && ax <= 1)
|
||||
return (cpackf(atanhf(x), y));
|
||||
|
||||
if (x == 0)
|
||||
return (cpackf(x, atanf(y)));
|
||||
|
||||
if (isnan(x) || isnan(y)) {
|
||||
if (isinf(x))
|
||||
return (cpackf(copysignf(0, x), y+y));
|
||||
if (isinf(y)) {
|
||||
return (cpackf(copysignf(0, x),
|
||||
copysignf(pio2_hi + pio2_lo, y)));
|
||||
}
|
||||
return (cpackf(x + 0.0L + (y + 0), x + 0.0L + (y + 0)));
|
||||
}
|
||||
|
||||
if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) {
|
||||
return (cpackf(real_part_reciprocal(x, y),
|
||||
copysignf(pio2_hi + pio2_lo, y)));
|
||||
}
|
||||
|
||||
if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) {
|
||||
raise_inexact();
|
||||
return (z);
|
||||
}
|
||||
|
||||
if (ax == 1 && ay < FLT_EPSILON)
|
||||
rx = (logf(ay) - m_ln2) / -2;
|
||||
else
|
||||
rx = log1pf(4 * ax / sum_squares(ax - 1, ay)) / 4;
|
||||
|
||||
if (ax == 1)
|
||||
ry = atan2f(2, -ay) / 2;
|
||||
else if (ay < FLT_EPSILON)
|
||||
ry = atan2f(2 * ay, (1 - ax) * (1 + ax)) / 2;
|
||||
else
|
||||
ry = atan2f(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2;
|
||||
|
||||
return (cpackf(copysignf(rx, x), copysignf(ry, y)));
|
||||
}
|
||||
|
||||
float complex
|
||||
catanf(float complex z)
|
||||
{
|
||||
float complex w = catanhf(cpackf(cimagf(z), crealf(z)));
|
||||
return (cpackf(cimagf(w), crealf(w)));
|
||||
}
|
||||
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Reference in a new issue